Question

Prove that there exists an infinite quantity of solutions to x^2=1in the quaternions. For example, in...

Prove that there exists an infinite quantity of solutions to x^2=1in the quaternions.
For example, in
H = { a + bi + cj +dk such that a,b,c,d∈R} and i^2 =j^2 =k^2 = ijk= −1
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Answer #1

We first show that the question is wrong. For, note that

since

Hence

which is true only when (since which is a contradiction) and

Thus, the equation in the quaternions has only two solutions .

Correct question: Prove that there exists an infinite quantity of solutions to in the quaternions.

Answer: From above we find that

which is true only when and hence we find that

Thus, the solution set of in the quaternions is given by

which has a bijective correspondance with the points on the unit sphere in . Hence there exists an infinite quantity of solutions to in the quaternions.

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